1. Math 3120 Differential Equations with Boundary Value Problems Chapter 2: First-Order Differential Equations Section 2-5: Solutions By Substitution

2. Homogenous Equations If a function f possesses the property for some real number, then f is said to be a homogenous functions of degree α. For example, A first-order DE in differential form is said to be homogenous if both coefficient functions M and N are homogenous equations of the same degree. Note: By using substitution ( y = ux or x =vy) we can reduce a homogenous equation to a separable first-order DE.

3. Method to solve Homogenous Equation Check if the DE is an homogenous equation. Write out the substitution y = ux Solve the new equation (which is always separable) to find u. Through the substitution go back to the old function y.

4. Example 1: Pg 74 Q2 Solve the given differential equation by using an appropriate substitution.

5. Example 2: Pg 74 Q4 Solve the given differential equation by using an appropriate substitution.

6. Example 3: Pg 74 Q11 Solve the given initial-value problem.

7. Bernoulli’s Equation The differential equation where n is any real number is called Bernoulli’s equation. Note For n=0 and n=1, equation (2) is linear. For n ≠ 0 and n ≠ 1, the substitution u = y 1-n reduces eq (2) to a linear equation.

8. Method to solve Bernoulli’s Equation Check if the DE is a Bernoulli’s equation. Find the parameter n from the differential equation. Write out the substitution u = y 1-n Solve the new linear equation to find v Through the substitution y = u (1/1-n) go back to the old function y.

9. Example 4: Solve the given differential equation by using an appropriate substitution.

10. Example 5: Solve the given initial-value problem.

11. Reduction To Separation Variables The differential equation can always be reduced to an equation with separable variables by means of the substitution u = Ax + By + C, B ≠ 0.

12. Example 6: Solve the given differential equation by using an appropriate substitution.